For the one-dimensional ideal gas, the velocity \(v(x,t)\) , mass density \(\rho (x,t)\)  and pressure \(p(x,t)\)  satisfy the Euler equations [1]}, [2]}
\(&\partial _t \rho + \partial _x (\rho v) = 0,\\&(\partial _t + v \partial _x)\ln (p/\rho ^3) = 0,\\&(\partial _t + v \partial _x) v = - \rho ^{-1}\partial _x p,\) 
